In mathematics, the Jacobson–Morozov theorem is the assertion that nilpotent elements in a semi-simple Lie algebra can be extended to sl2-triples. The theorem is named after Jacobson 1951, Morozov 1942.
Statement
The statement of Jacobson–Morozov relies on the following preliminary notions: an sl2-triple in a semi-simple Lie algebra 
(throughout in this article, over a field of characteristic zero) is a homomorphism of Lie algebras 
. Equivalently, it is a triple 
of elements in satisfying the relations
An element 
is called nilpotent, if the endomorphism 
(known as the adjoint representation) is a nilpotent endomorphism. It is an elementary fact that for any sl2-triple 
, e must be nilpotent. The Jacobson–Morozov theorem states that, conversely, any nilpotent non-zero element 
can be extended to an sl2-triple. For 
, the sl2-triples obtained in this way are made explicit in Chriss & Ginzburg (1997, p. 184).
The theorem can also be stated for linear algebraic groups (again over a field k of characteristic zero): any morphism (of algebraic groups) from the additive group 
to a reductive group H factors through the embedding
Furthermore, any two such factorizations
are conjugate by a k-point of H.
Generalization
A far-reaching generalization of the theorem as formulated above can be stated as follows: the inclusion of pro-reductive groups into all linear algebraic groups, where morphisms 
in both categories are taken up to conjugation by elements in 
, admits a left adjoint, the so-called pro-reductive envelope. This left adjoint sends the additive group to 
(which happens to be semi-simple, as opposed to pro-reductive), thereby recovering the above form of Jacobson–Morozov.
This generalized Jacobson–Morozov theorem was proven by André & Kahn (2002, Theorem 19.3.1) by appealing to methods related to Tannakian categories and by O'Sullivan (2010) by more geometric methods.